{"paper":{"title":"The set of dimensions for which there are no linear perfect 2-error-correcting Lee codes has positive density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Claudio Qureshi","submitted_at":"2018-04-24T23:13:07Z","abstract_excerpt":"The Golomb-Welch conjecture states that there are no perfect $e$-error-correcting Lee codes in $\\mathbb{Z}^n$ ($PL(n,e)$-codes) whenever $n\\geq 3$ and $e\\geq 2$. A special case of this conjecture is when $e=2$. In a recent paper of A. Campello, S. Costa and the author of this paper, it is proved that the set $\\mathcal{N}$ of dimensions $n\\geq 3$ for which there are no linear $PL(n,2)$-codes is infinite and $\\#\\{n \\in \\mathcal{N}: n\\leq x\\} \\geq \\frac{x}{3\\ln(x)/2} (1+o(1))$. In this paper we present a simple and elementary argument which allows to improve the above result to $\\#\\{n \\in \\mathca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.09290","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}